A q-CONTINUED FRACTION
نویسندگان
چکیده
Let a, b, c, d be complex numbers with d 6= 0 and |q| < 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq + cq (a + b)qn+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d)(−c/bd)j q (q)j(−aq/d)j P∞ j=0 (b/d)(−c/bd)j q (q)j(−aq/d)j . We then use this result to deduce various corollaries, including the following: 1 1 − q 1 + q − q 1 + q2 − q 1 + q3 − · · · − q2n−1 1 + qn − · · · = (q; q)∞ (q; q)∞ ,
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